One-Shot Games for Spectrum Sharing among Co-Located Radio Access Networks

Transcription

1 One-Shot Games for Spectrum Sharng among Co-Located Rado Access etwors Sofonas Halu, Alexs A. Dowhuszo, Olav Tronen and Lu We Department of Communcatons and etworng, Aalto Unversty, P.O. Box 3000, FI Aalto, Fnland Department of Mathematcs and Statstcs, Unversty of Helsn, P.O. Box 68, FI-0004, Fnland E-mal: {sofonas.halu, alexs.dowhuszo, Abstract We consder an adaptve co-prmary spectrum sharng scheme based on playng one-shot games between co-located dstrbuted Rado Access etwors (RAs). Spectrum s dvded nto prvate and shared parts, and each RA nternally apples coordnated mult-pont transmssons. Zero-forcng precodng s used n the prvate part, whle untary precodng s appled n the non-orthogonally shared part. A one-shot game s played such that each operator proposes a spectrum partton whch maxmzes ts own sum utlty, and the actual spectrum partton s resolved based on a pror decson rules that reflect the regulatory framewor. Thus, adaptve spectrum sharng can be mplemented wth a mnmal exchange of sgnalng nformaton among operators. The exstence of a unque ash-equlbrum s shown for the game. Smulaton results show that adaptve spectrum sharng s able to brdge between the baselne performance of orthogonal and full (non-orthogonal) spectrum n dfferent sgnal-to-nose power rato regons. I. ITRODUCTIO In order to cope wth the forecast demand of moble data traffc n future wreless networs, operators need to sgnfcantly ncrease ther operatonal bandwdth []. Inter-operator spectrum sharng s one of the approaches that has been envsoned to ncrease the operatonal bandwdth of an operator. The conventonal cogntve rado networ approach that s used to share spectrum between operators assumes a pre-defned prmary secondary herarchy. In an alternatve co-prmary spectrum sharng model, multple operators are wllng to jontly use a part (or the whole) of ther lcensed spectrum [], [3]. Operators may have ndvdual lcenses to access exclusve frequency bands, or a group authorzaton to use a common pool of spectral resources. Jont use of the lcensed spectrum of operators can be realzed by sharng the frequency resources ether orthogonally or non-orthogonally. Orthogonal sharng of common spectral resources could be acheved n tme doman [4], frequency & power doman [5], [6], and/or space doman [7]. In the case of non-orthogonal spectrum sharng, operators smultaneously use common bloc of spectral resources, creatng nter-operator nterference. The operators mght mplement nter-operator nterference mtgaton schemes, see e.g. [8], [9]. Most of the wors on co-prmary spectrum sharng rely on operator specfc nformaton whch n some cases needs to be exchanged. In [8], [9], operator specfc nformaton such as full nter-operator Channel State Informaton (CSI) s assumed to be avalable for all operators, n order to apply nter-operator nterference mtgaton schemes. However, the exchange of nter-operator CSI generally requres a dedcated ln between the operators or to a central entty. In [5], [6], game theoretc approaches ncludng one-shot games are dscussed for unlcensed spectrum sharng, where the games depend on the power appled on all carrers by the operators. Wth these approaches, the rate that an operator could acheve on a gven carrer or porton of the spectrum pool s unpredctable, as the nter-operator nterference depends on the power appled on the carrer by other operators. In addton, both [5] and [6] consdered systems comprsed of sngle transmtter-recever pars leavng no room for mult-user dversty gan. On the other hand, n [0], a statc co-prmary spectrum sharng scheme s proposed, whch restrcts the gan from flexbly sharng the spectrum though t mght be easer to mplement. In ths paper, a one-shot game approach s consdered for co-prmary spectrum sharng among co-located dstrbuted Rado Access etwors (RAs) of dfferent operators. The operators do not exchange CSI, only preferences of parttonng the spectrum nto prvate and non-orthogonally shared portons are exchanges. The decson to determne the actual spectrum partton s made based on a pror decson rule, tang nto account the regulatory framewor. The proposed one-shot game s shown to have a unque ash equlbrum. Thus, the game could be played once per nterval of tme n whch the channel realzaton of all the users reman constant. Smulaton results ndcate that the proposed one-shot game leads to better and more far data rate allocatons to the users, when compared to conventonal schemes that share the whole communcaton bandwdth non-orthogonally among operators, or allocate an exclusve frequency band to each of the operator. The rest of the paper s organzed as follows: Secton II explans the system model and presents the spectrum access schemes. Secton III dscusses the Coordnated Mult-Pont Transmsson (CoMP) technques appled n the prvate and non-orthogonally shared portons of the spectrum, and descrbes the computaton of the sum utlty of the operators. Secton IV presents the proposed one-shot game and proves the exstence of a unque ash-equlbrum soluton. Secton V dscusses the characterstcs of the smulaton scenaro and carres out performance analyss. Fnally, conclusons are drawn n Secton VI.

2 Operator RAU MS BBU Prvate Band for operator Shared Band Prvate Band for operator Operator RAU MS BBU Fg.. Mult-operator downln communcaton scenaro wth I = dstrbuted RAs, coverng an overlappng hot-spot area. Total communcaton spectrum s dvded nto three parts: a prvate band for each operator and a non-orthogonally shared band. The spectrum parttonng between operators s coordnated usng control sgnalng between both BBUs of the operators. II. SYSTEM MODEL A. Mult-operator downln scenaro A mult-operator downln communcaton scenaro s consdered whch conssts of co-located multple operators belongng to the set I, see llustraton for I = n Fg.. The operators own ndependent dstrbuted Rado Access etwors (RAs), ther coverage area are overlappng, and they are wllng to share ther lcensed spectrum to mprove the performance of ther Moble Statons (MSs). The dstrbuted RA of operator I conssts of M Remote Antenna Unts (RAUs) wth wred connectons to a centralzed Baseband Unt (BBU). The centralzed BBU of operator performs schedulng decsons and carres out the mult-antenna sgnal processng operatons that are requred to serve K randomly deployed sngle-antenna MSs n the coverage area. For sae of smplcty, we assume that the number of RAUs and MSs per operator are equal,.e., M = K I. It s assume that the users are relatvely statc. Accordng to spectrum access model used, operator may propose to share non-orthogonally part of ts dedcated ndvdual spectrum B nd, or may request to have exclusve access to a porton of the common group spectrum B grp. egotatons dealng wth the most convenent spectrum partton are carred out among operators n a tme scale longer than schedulng nterval, but shorter than the coherence tme of channels and the traffc needs of the users. The results of a negotaton holds for one sharng perod. The negotaton are based on a mnmal exchange of control nformaton. Operators do not share local nformaton le schedulng decsons and transmt beamformng vectors, the only control nformaton that s shared s the most convenent spectrum partton from the perspectve of each ndvdual operator. The decson on the spectrum partton used for the next sharng perod s carred out by each operator n a decentralzed way, followng an a pror decson rule that all operators are forced to follow. Accordng to the reached decson, for the duraton of the next sharng perod, shared sub-band B (s) s used for non-orthogonal spectrum access among all operators, whle prvate sub-band B (p) s used for orthogonal spectrum access of operator. Then, the total communcaton bandwdth for the mult-operator scenaro s B = B (s) + I B(p), and the aggregate communcaton bandwdth that operator uses to serve ts K MSs s B = B (p) + B (s). The transmt power spectral densty of each operator s assumed to be constant across ts aggregate bandwdth B. Thus, P tx, /B = tx, /B(s) = P (p) tx, /B(p), where P tx, = P (p) tx, + tx, s the total transmt power that operator uses for communcaton, whle P (p) (s) tx, and P tx, are the correspondng transmt power portons n the prvate and shared sub-bands, respectvely. ote that based on ths assumpton, transmt power P tx, s a lnear functon of bandwdth B. The wreless channel gan between RAU m of operator and MS s gven by h (m,), = h (m,), / L (m,), I, K, () where L (m,), s the dstance dependent pathloss attenuaton and h (m,), are the complex fast fadng components of the wreless channel model. In case of rch scatterng and flat fadng, h (m,), s descrbed by a zero-mean crcularly symmetrc complex Gaussan Random Varable (RV) wth unt varance. All fast fadng components of the wreless channel are assumed to be uncorrelated. Fnally, the nose powers that MS experences n prvate and shared sub-bands are modeled as a zero-mean Gaussan complex RVs wth power = 0 B (s) and P (p), = 0 B (p), wth 0 the thermal spectral densty. B. Spectrum sharng models The negotatons to determne the spectrum partton for the whole mult-operator scenaro are performed n a decentralzed manner between peers. In ths process, each operator frst nforms to the other operators ts preferred spectrum partton. ext, the actual spectrum partton s resolved locally by each operator, based on pre-defned rules that tae nto account the type of lcense that operators have to access spectrum. Based on the spectrum lcense of the operators, two co-prmary sharng models are consdered []. ) Mutual rentng: Each operator owns a lcense for exclusve access to a frequency band of bandwdth B nd. For smplcty, we assume equal amounts of lcensed spectrum,.e, B nd = B nd I. Operator determnes ts preferred fracton 0 α nd that t s wllng to share and nforms ths to the other operators. If proposal α nd s accepted by all operators, the total bandwdth of the shared sub-band becomes B (s) = I α nd B nd, whle the bandwdth of the prvate sub-band s reduced to B (p) = B nd B (s) / I I.

3 ) Lmted spectrum pool: In ths case, the I operators have a lcense to access a pool of common frequency resources wth total bandwdth B grp, whch s assumed to be non-orthogonally shared among all operators. Then, operator determnes ts preferred fracton 0 α grp / I that t prefers to use prvately. If proposal α grp s accepted by all operators, the bandwdth of the prvate sub-band becomes B (p) = α grp B grp I, whle the total bandwdth of the shared sub-band s reduced to B (s) = B grp I B (p). III. UTILITY FUCTIO AD MULTI-ATEA TRASMISSIO SCHEMES For a gven spectrum partton {B (p), B (s) }, operator has the opportunty to schedule ts assocated MSs ether n the prvate, shared, or both prvate and shared frequency sub-bands. In the shared frequency sub-band B (s), full-ran Untary Precodng (UP) s used to mae the nstantaneous nter-operator nterference power reman constant, ndependently of the local schedulng decsons that operators mae. In the prvate frequency sub-band B (p), on the other hand, Zero-Forcng Precodng (ZF) s employed to cancel completely the ntra-operator nterference that s created among the data streams ntended to the dfferent scheduled users. Let K be the set of MSs assocated to operator. Then, the ndexes of those MSs that are scheduled n the prvate and shared frequency sub-bands are contaned n sets S (p) K and S (s) K, respectvely. In ths paper, S (p) S (s) = K s always verfed. A. Receved SIR n prvate and shared frequency sub-bands The data rate that a MS K s able to acheve depends on both the spectrum partton {B (p), B (s) } and the schedulng decsons {S (p), S (s) }, where the latter affects the Sgnal-to-Interference plus ose power Rato (SIR) that the MS observes n recepton. Then, the SIR that MS K experences n the shared frequency sub-band s γ (s) = P l S (s),l (s) tx, ht,w (s) tx, ht, w(s), +,l } {{ } Intra-operator nterference l S (s) j,j tx,j ht j,w (s) j,l } {{ } Inter-operator nterference, + }{{} ose () where h T, s a row vector that contans all channel gans from RAUs of operator to MS, w (s), s the transmt beamformng vector that operator uses to serve MS n the shared frequency sub-band, and s the nose power n the shared frequency sub-band. ote that when MS / S (s), beamformng vector w (s), s a null vector of the proper dmenson. Smlarly, the SIR that MS K experences n recepton n the prvate frequency sub-band s obtaned from () when the superscrpts (s) are replaced wth (p), and when the nter-operator nterference term of the denomnator s removed. ote that f ZF s used to serve MSs n the prvate frequency sub-band, the ntra-operator nterference term that appears n the denomnator of () vanshes as well. Algorthm Untary precoder for shared frequency sub-band : Intalzaton: Set U = S (s) and D 0 = : for m = to S (s) do 3: for n = to U m do 4: v n (m) h H,n / h,n d D u T d (hh,n / h,n )u d 5: v n (m) v n (m) /( v n (m) ) 6: I (m) n 7: γ (m) n = S (s) = d D tx, ht,n u d I (m) n 8: end for 9: π m = arg mn n Um γ n (m) 0: D m = D m {π m } : U m+ = U m {π m } : u m = v π (m) m tx, ht,n v(m) n + l S (s) (s) P,j j tx,j ht j,nw(s) j,l + 3: end for 4: Output: W (s) = [ ] (s) u u (s) S and D = D (s) S B. Sum utlty functon The operator sum utlty s defned as U = K u (r ) I, (3) where u (x) = log e (x) n case of a proportonal far log-rate utlty, and r = B (p) log ( + γ (p) )+B(s) log ( + γ (s) ) K (4) s the aggregate rate that MS s able to acheve. ote that Aggregate rate (4) depends on the bandwdth of spectral portons B (p) and B (s), as well as the receved SIRs γ (p) and γ (s) that MS experences n both prvate and shared frequency sub-bands, respectvely. C. Untary precoder n shared frequency sub-band The columns of the untary precoder matrx W (s) that s used to serve MSs S (s) are determned by an orthogonalzaton process. The order of orthogonalzaton depends on the receved SIR γ (m) that each MS experences n recepton at each teraton m of the process, and taes nto account both ntra-operator and nter-operator nterference powers. The ntra-operator nterference power comes from the MSs that were orthogonalzed n a prevous teraton of the process, whle the nter-operator nterference power remans constant durng the whole process because t s assumed that the other operators also use UP n the shared frequency sub-band. To ncrease the level of farness among users, the MS wth lowest SIR s selected for orthogonalzaton at each teraton. A summary of the procedure that s used to determne the untary precoder for the shared frequency sub-band of operator s presented as Algorthm.

4 D. Zero-forcng precoder n prvate frequency sub-band The MSs scheduled n the prvate frequency sub-band are served usng ZF wth sum power constrant. The precoder matrx that operator uses n the prvate frequency sub-band becomes W (p) where H +,S (p) H (p),s h T, = [ w (p), w(p), S (p) ] = H +,S (p) H +,S (p) F I, (5) s the pseudonverse of the channel matrx, whch s formed by the concatenaton of row vectors S(p), and F s the Frobenus norm of a matrx. ote that the pseudonverse of matrx H s defned as H + = (H H H) H H where ( ) and ( ) H are the nverse and Hermtan transpose of a matrx, respectvely. IV. PROPOSED OE-SHOT GAME In ths paper, we model the co-prmary spectrum sharng problem as a one-shot game { } I, {a } I, {U } I. In ths defnton, I denotes the set of players the operators wth servng the same hotspot area. The set {a } I denotes the nstantaneous strateges of the players, and the set {U } I dentfes ther correspondng payoffs. The allowed strategy of a player s restrcted to an nterval a [0, α max ] I, and have dfferent meanng dependng on the co-prmary spectrum sharng model. In the case of mutual rentng, the strategy of a player represents the fracton of ts lcensed spectrum that the operator s wllng to share wth the other operators, and α max = f B Ind = Bj Ind, j I. In the case of lmted spectrum pool, the strategy of a player denotes the fracton of common spectrum resource that the operator s requestng for prvate (exclusve) use, and α max = / I. The payoff U (a, a ) of player s defned as the sum of the utltes of ts assocated MSs after the proposals that dfferent players proposed for parttonng the spectrum are resolved. Accordng to ths defnton, the payoff of a player depends not only on ts own strategy a, but also on the strategy of the other players a and the a pror rule that all players agree to follow n order to resolve ther spectrum partton proposals. Tang nto account the spectrum regulatory framewor, we propose an a pror rule to resolve the outcomes of the spectrum parttonng proposals by and thus U (a, a ) = U (a mn ), a mn = mn I a, (6) a mn = mn I a. (7) Therefore, n the case of mutual rentng, an operator s guaranteed that t s not gong to share more than the fracton of ts lcensed spectrum that t s wllng to share. Smlarly, n the case of lmted spectrum pool, an operator s guaranteed that t wll not be forced to non-orthogonally share less than the fracton of the common resource that t s wllng to share. A. Selectng the strategy of a player For smplcty, we requre that each user s always scheduled n both prvate and shared frequency sub-bands. Then, t s possble to show that the utlty functon of an operator s strct concave functon of a f the log-rate utlty functon s used. Ths follows from the observaton that, when schedulng decsons are fxed, the data rate of an MS s a lnear functon of a, and t s well-nown that the sum of logarthms of lnear functons s a strctly concave functon []. A player chooses a strategy that maxmzes ts payoff functon. However, the payoff functon of a player depends on the nstantaneous strateges of all the players, and a player cannot now n advance the strateges of other players. As the CSIs of the other players are not nown, n a generc game, a player would have to apply a probablty dstrbuton of all the strateges of the other players to deduce ts chosen strategy, so that the outcome of the game would be closest to the preferred one. The outcome decson rule consdered here, however, together wth concavty, smplfes the strategy selecton process. We assume that player selects ts strategy wthout consderng probablty dstrbutons of other player s strateges, a = arg max a U (a ) subject to 0 a α max S (p) = S (s) = K, (8) ote that n (8), the sum utlty for a partcular a [0, α max ] s obtaned computng the correspondng B (p) and B (s) frst, and then pluggng the obtaned results n equaton (4) and (3). Due to the concavty of U (a ), and the outcome decson rule (6), ths s the optmal strategy choce for, rrespectvely of the other players actons. Ths wll be seen below, and s the reason for the exstence of a ash equlbrum. The soluton to (8) can be readly obtaned usng effcent convex optmzaton algorthms, le the ones presented n []. B. Exstence and unqueness of ash-equlbrum A unque ash-equlbrum exsts for the proposed one-shot game, when players are ratonal and try to selfshly maxmze ther own sum utlty. Proposton. The proposed one-shot game has a unque ash-equlbrum pont a = {a,..., a,..., a I } where a, I s chosen accordng to (8), the payoff U (a, a ), I s gven by (7), and the sum utlty U (a ) n (8) s a strct concave functon of a. Proof. Consder a player unlaterally changng ts strategy from a to a, and let a,mn = mn{a j } j,j I. We now analyze two complementary cases: Case. Assume that a >= a,mn If a a,mn, U(a, a ) = U(a, a ). If a < a,mn, U(a, a ) < U(a, a ) snce U(a, a ) s a strct concave functon n a [0, a,mn ).

5 Case. Assume that a < a,mn For a a, U(a, a ) < U(a, a ). Thus a ash equlbrum exsts. The unqueness of the ash-equlbrum follows from the strct concavty of U (a ) n (8), snce t s a sum of log functons. Thus, the acton of a player whch s selected by maxmzng (8) s unque. It s worth to notce, that the game settng dscussed here allows for a ash equlbrum where each operators do not always use the full spectrum. Ths s n contrast to the spectrum sharng games n frequency/power doman [5], where the ash equlbrum s to use the full frequency. The dfference n outcomes s due to Rule (6) of the game consdered here. In the game model here, (6) s outsde the doman of the decson of the players, and enables non-trval outcomes of the game. V. PERFORMACE RESULTS To understand characterstcs of the consdered spectrum sharng game, we smulate t n a mult-operator downln scenaro. A. Smulaton Scenaro We consder I = co-located dstrbuted RAs belongng to dfferent operators. We use an evaluaton scenaro smulaton smlar to the small cell networ deployments model of []. The networ elements of both RAs are ndependently deployed to serve the same hotspot area. Each operator unformly deploys a total of M = 4 RAUs n an nner crcle of radus R, and a total of K = 4 sngle-antenna MSs n an outer concentrc crcle of radus R. Mnmum dstances d mn,rau-rau = 0 m and d mn,rau-ms = 3 m are respected when the networ elements are owned by the same operator. There s no mnmum RAU-to-RAU or RAU-to-MS dstance restrcton when the networ elements belong to dfferent operators. The dstance dependent path loss attenuaton s calculated usng the Urban Mcro (UM) wreless channel model wth a carrer frequency of 3.4 GHz [3]. For sae of smplcty, we assume that there s always a on Lne-of-Sght (LOS) condton between each RAU and MS par; therefore, the path loss attenuaton s L (m,), = 36.7 log 0 (d (m,), ) log 0 (f c ), (9) where d (m,), s the dstance n meters between RAU m of operator and MS, and f c s the carrer frequency n GHz. The antenna gans are G rau = 5 db and G ms = 0 db for the RAUs and MSs, respectvely. The nose fgure of the MSs s set to F ms = 9 db. For the mutual rentng regulatory model, both operators are assumed to have exclusve access to a frequency band of bandwdth B nd = 0 MHz I. Smlarly, n the lmted spectrum pool regulatory model, both operators are assumed to have authorzaton to access a common pool of frequency resources of bandwdth B grp = 0 MHz. The transmt power spectral densty s 0 dbm/mhz, and s ept constant ndependently on the aggregate communcaton bandwdth B. Average Data Rate per User (Mbps) Adaptve Spectrum Sharng (LSP) Adaptve Spectrum Sharng (MR) Orthogonal Spectrum Sharng on orthogonal Spectrum Sharng Radus of nner crcle (m) Fg.. Mean data rate per user for dfferent spectrum sharng schemes as a functon of nner crcle radus R (outer crcle radus R =.4 R ). Dashed lnes: Fxed spectrum sharng schemes. Sold lnes: Adaptve spectrum sharng schemes. MR: Mutual rentng. LSP: Lmted spectrum pool. 0 th Percentle of User Data Rates (Mbps) Adaptve Spectrum Sharng (LSP) Adaptve Spectrum Sharng (MR) Orthogonal Spectrum Sharng on orthogonal Spectrum Sharng Radus of nner crcle (m) Fg th percentle of user data rates for dfferent spectrum sharng schemes as a functon of nner crcle radus R (outer crcle: R =.4 R ). Dashed lnes: Fxed spectrum sharng schemes. Sold lnes: Adaptve spectrum sharng schemes. MR: Mutual rentng. LSP: Lmted spectrum pool. B. Analyss of numercal results In order to study the effect that the Sgnal-to-ose power Rato (SR) has on the performance of the proposed spectrum sharng schemes, the radus of nner crcle R (.e., the deployment area for RAUs) and the radus of the outer crcle R (.e., the deployment area for MSs) are vared eepng the rato R /R =.4. Ths s n lne wth the baselne recommendaton presented n [], whch defnes R = 50 m and R = 70 m. For each value that R taes, numercal smulatons are run for 0 ndependent RAU deployments, each of them havng 0 ndependent deployments of MSs. For each separate RAU and MS snapshot, 0 ndependent fast fadng states are used when modelng the gans of the wreless channel. Orthogonal spectrum sharng s used as baselne scheme for performance evaluaton for mutual rentng scenaro. When operators have a group lcense to access a spectrum pool, full non-orthogonal spectrum sharng s the baselne. Fgure shows the mean data rate per user (.e., from an average rate performance perspectve), and Fgure 3 presents the 0-th percentle of user data rate, whch reflects outage rate performance. Accordng to smulaton results presented n Fgure and

6 Fgure 3, orthogonal spectrum sharng wors well n the hgh SR regon, whle full spectrum sharng wors better n the low SR regon,.e., when R grows and the mult-operator scenaro becomes nose-lmted. Adaptve spectrum sharng wth one shot game protocols provdes a performance that s at least as good as the one that s acheved wth the fxed spectrum allocaton schemes prevously descrbed. Ths observaton s vald for both mean and 0-th percentle data rate curves. For md-range SR values,.e. for R n the range of m, the gan of adaptve spectrum sharng s notable when compared to both baselne spectrum sharng schemes, partcularly when the 0-th percentle data rate fgure s analyzed n detal. Fnally, for md-range SR values, adaptve spectrum sharng wth group lcense and lmted spectrum pool wors slghtly better n terms of mean user data rate, whle adaptve spectrum sharng wth ndvdual lcense and mutual rentng has a better performance n terms of the 0-th percentle of the user data rate. VI. COCLUSIO In ths paper, a one-shot game was proposed to mplement adaptve co-prmary spectrum sharng among co-located dstrbuted RAs of dfferent operators. The game was consdered n dfferent forms for an ndvdual lcensng wth mutual rentng and group lcense model wth lmted spectrum pool, and can be mplemented wth a mnmal exchange of control sgnalng. The game was shown to have a unque ash equlbrum n all networ states. The performance of the proposed adaptve spectrum sharng scheme was analyzed n terms of both mean user rate and 0-th percentle of user rate, assumng a dense small cell networ scenaro wth multple operators servng the same hotspot area. For low and hgh SR regons, adaptve spectrum sharng scheme s able to choose the better of full spectrum sharng and orthogonal spectrum sharng. For md-range SR values, adaptve spectrum sharng scheme showed a notable gan wth respect to both fxed spectrum sharng schemes. It can be concluded that adaptve spectrum sharng based on mnmal nformaton exchange between operators holds promse for guaranteeng more spectrum and correspondng better servce for users served by future communcaton systems. ACKOWLEDGMET Ths wor was supported n part by the Academy of Fnland (grant 5499), and the European Commsson n the framewor of the FP7 project ITC METIS. REFERECES [] oa Semens etwors, Wae-up call: Industry collaboraton needed to mae beyond 4G networs carry 000 tmes more traffc by 00, Whte Paper, Aug. 0. [] T. Irnch, J. Kronander, Y. Selén, and G. L, Spectrum sharng scenaros and resultng techncal requrements for 5G systems, n Proc. IEEE Int. Symp. Personal, Indoor and Moble Rado Commun., Sept. 03, pp [3] METIS 00 Project, Intermedate descrpton of the spectrum needs and usage prncples, Delverable D5., Aug. 03. [4] G. Mddleton, K. Hool, A. Töll, and J. Llleberg, Inter-operator spectrum sharng n a broadband cellular networ, n Proc. IEEE Int. Symp. Spread Spectrum Technques and Applcatons, Aug. 006, pp [5] R. Etn, A. Pareh, and D. Tse, Spectrum sharng for unlcensed bands, IEEE J. Sel. Areas Commun., vol. 5, no. 3, pp , Apr [6] M. Benns, S. Lasaulce, and M. Debbah, Inter-operator spectrum sharng from a game theoretcal perspectve, EURASIP J. Adv. Sgnal Process., vol. 009, pp., Sept [7] E. A. Jorswec, L. Bada, T. Fahldec, E. Karpds, and J. Luo, Spectrum sharng mproves the networ effcency for cellular operators, IEEE Commun. Mag., vol. 5, no. 3, pp. 9 36, Mar. 04. [8] R. Gangula, D. Gesbert, J. Lndblom, and E. G. Larsson, On the value of spectrum sharng among operators n multcell networs, n Proc. IEEE Veh. Tech. Conf., Jun. 03, pp. 5. [9] S. Halu, A. A. Dowhuszo, and O. Tronen, Adaptve co-prmary shared access between co-located rado access networs, n Proc. Int. Conf. Cogntve Rado Orented Wreless etw., Jun. 04, pp [0] Y. Teng, Y. Wang, and K. Horneman, Co-Prmary spectrum sharng for denser networs n local area, n Proc. Int. Conf. Cogntve Rado Orented Wreless etw., Jun. 04, pp [] S. Boyd and L. Vandenberghe, Convex optmzaton. Cambrdge unversty press, 009. [] 3GPP TSG RA WG, Evaluaton assumptons for small cell enhancements - physcal layer, Tech. Doc. R-30856, Jan. 03. [3] 3GPP, Further advancements for E-UTRA physcal layer aspects (Release 9), Tech. Rep. TR V9.0.0, March 00.